From your avatar I am assuming you have canaries? Please let me know if that isn't the case. My finches, especially Zippy, get time to free fly my house. After Peppy, another finch, joined us the home dynamic changed. Zippy would fly out and hop around calling to Peppy, but Peppy refused to leave so Zippy would give up and return to the cage. Lately they don't want to leave at all. It makes it complicated when you want to clean the cage well. I ended up placing them in a travel carrier so that I could wash the sides of their cage well and change perches. Luckily the lower portion of the cage has a pull out tray so that is easier. I needed to carrel them into the carrier, delicately. Zippy is pretty human curious but doesn't like to be touched, though I did need to cup him with my hands.
When Zippy was free flying more I used food as a way to return him to the cage. He would fly about for about an hour, then he would get tired and hungry and return for a bite. If you want your canaries to exit, without needing to place them in another carrier, I would place their food and water on a surface that they can see from the cage. They will likely fly out for food and then you can clean the cage. However since they aren't used to flying around your home, they will likely be a bit fearful of the new setting. Make sure you bird-proof; scared birds can get stuck behind things, fly into mirrors and windows, or crash into walls. When you wait for them to return to the cage, using the same food method, it may take hours. If I were you I would clean them on a day when you can stay home and have no commitments, so you can ensure supervision. If it goes well you can start letting them out more often and they will get more confident in the home. However like you said, they may not want to leave.
Edit in response to budgies:
I would say that for wall scrubbing or perch replacing it is a good idea to move canaries or finches from the immediate environment. They are very fragile and will panic when things are moving and can get crush. I personally always move them for major cleans. They can stay in the cage for daily cleaning like the tray, water dishes and the like.
